12 UNBOUNDED NORMAL OPERATORS 419 (15.12.8.1) Let (En) be a sequence of closed subspaces ofE, such that E is the Hilbert sum of the En . For each n, let Tn be a continuous operator on En . Then there exists a unique closed operator T on E such that, for all n, En cz dom(r), T| En = Tn , PET -x = TPEn<x (xe dom(T)). Moreover, dom(T) to the set F 0/ #//* = £ xn (with xn e Enfor all n) such that E \\Tn ' *J2 < +00, and we have T-x = ^Tn-xn. n n Since we must have £ \\PEn - (T • x)\\2 = || T- x\\2 < + oo, we see that the n conditions stated imply that dom(T) c F and that T - x = £ Tn • xn for all n x = Z xn e dom(T). Note that F is a vector subspace of E, by virtue of the inequality \\z + z'\\2 g 2(\\z\\2 + \\z'\\2) in E. For each x = %xneF, write n T • x = £ rn * *„; we shall show that the operator T : F-» E so defined is n closed. Since clearly F(T) contains all finite sums ]T (JCM , Tn * jtn), it follows that n F(T) is dense in r(T"), and since T is a closed operator, this implies that T= T'. It is clear that T' is linear. Also, if a sequence (xm) in F tends to x e E and if the sequence ( T' • xm) tends to y e E, then PEn T' • xm = TnPEn - xm tends to PEn • y and also to TnPEn • x, by virtue of the continuity of Tn; this proves that x e F and that y = T • x, and completes the proof of the lemma. This lemma, applied to the situation of (ii), proves the uniqueness of N9 which therefore has to be the closed operator (15.12.7) described in the state- ment of (15.12.8(ii)). It remains to prove that N is normal. For each x e F, we have \\N*PEn - x\\ = \\NnPEn • x\\9 and therefore the lemma (15.12.8.1), applied to the family of continuous operators (N *), shows that there exists a unique closed operator N' such that, for all 72, we have En cz dom(7Vr/), N' \ En = N* , andPEn N' • x = NfPEn • x for all x e dom(N'). Moreover, we have dom(N') = dom(AO = F> a^d if "* = £*» 6 F, then N' - x = £ N* • xn . From these n n formulas it follows immediately that (N • x\y) = (x\Nf - y) for all x, y in F: in other words (15.12.3), F is contained in dom(7V*), and N' = N* \ F. But also PEnN* - y = N*PEn - y for all 77, and all y e F. For if xn e En , then (xn | P^N* " X> = fe | N* - y) (1 5.5.4), and since xn e dom(JV) because N- xn = A^n * xn eEn; hence finally, since En e dom(A^*), we have (*J-PEnN* *j) = (xn|7V*PEn - y), which proves our assertion. Since N* is closed (15.12.4), it follows that Nf = ^V* by virtue of (15.12.8.1). The preced-ion principle" is satisfied,